golden ratio essay - community college of rhode islandfaculty.ccri.edu/joallen/m1470/sample...

Madison Bozinoff Thursday, May 14th, 2015 Integrative Seminar 2: Visual Culture Final Research Paper Margot Bouman

How has the knowledge on the relation between math and nature had an

influence and evolved architectural design throughout history?

Madison Bozinoff

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Table of Contents

The Golden Ratio in Ancient Greek Architecture………………………………………… Page 3

Fibonacci Solidifying the Connection Between Math and Nature……………………….. Page 5

Fetchner and the Golden Rectangle………………………………………………………. Page 8

A Contemporary Example of the Golden Rectangle in Architecture…………………….. Page 9

Figures Cited…………………………………………………………………………….. Page 11

Works Cited……………………………………………………………………………… Page 15

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How has the knowledge on the relation between math and nature had an influence and

evolved architectural design throughout history?

Throughout history the relation between mathematics and nature has had an influence on

architecture. Various events in the history of mathematics, have led to new discoveries of these

relations between math and nature which have, in turn, had an effect on aesthetics and

architectural design at different stages in history.

The Golden Ratio in Ancient Greek Architecture:

The conversation about the relation between nature and mathematics begins in Ancient

Greece with three of the Ancient worlds greatest minds; Phidias, Plato and Euclid. Phidias most

famously designed the Parthenon in Athens, Greece in 447 BC . It is no surprise that the 1

mathematical theory began with Ancient Greece as mathematics and geometry were regarded as

sacred at that time. Phidias is believed to have lived from around 490 BC to 430 BC. Though it

is truly unknown where the golden ratio was first discovered, it was first studied by Phidias

while he was designing the Parthenon . The golden ratio is marked as 1:1.618. The influence of 2

his studies is visible in the construction and proportions of the Parthenon . 3

Gary Meisner, a journalist with extensive knowledge and publications on the golden

ratio, analyzed the proportions of the Parthenon with the PhiMatrix software, a software that

"Ancient Greece: The Parthenon." British Museum. Accessed April 12, 2015. https://www.britishmuseum.org/PDF/1Visit_Greece_Parthenon_KS2.pdf.

"Phidias." Ancient Greece. 2012. Accessed May 13, 2015. http://www.ancientgreece.com/s/People/Phidias/.2

H. E. Huntley. The Divine Proportion: A Study in Mathematical Beauty,. New York, New York: Dover Publications, 1970.3

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draws golden ratio proportions. He found that there are many instances were these proportions

come up in the design of the Parthenon. For example, the ratio of the height of the Parthenon to

the height of the columns to the height of the roof is 1:1.618. There is also the repetition of this

ratio in the height of the dividing section between the roof and the columns and the triangular

section of the roof. These are illustrated in Figure 1. The golden ratio is also present within the

dividing structure between the roof and columns. We can see in Figure 2 that the ratio is present

both between the two sections within the dividing structure, represented by the red rectangles, as

well as in the spacing out of the miniature columns that are in set out in sets of three, represented

by the yellow rectangles . 4

Phidias’ use of and research into the golden ratio gave rise the naming of the sacred

number in the golden ratio: 1.618 as Phi, the first syllable in Phidias’ name . 5

Another early supporter of the belief of the golden ratio in Ancient Greece was Plato.

Plato firmly believed that mathematics was the imperturbable foundation of all organic forms.

He believed that this form of mathematics was the “architecture of the universe” in a sense . 6

Meisner notes that in Plato’s Timaeus, Plato views the golden ratio “to be the most binding of all

mathematical relationships and the key to the physics of the cosmos” . In platonic geometry we 7

Gary Meisner. "The Parthenon and Phi, the Golden Ratio." TheGoldenNumber.net. January 20, 2013. Accessed May 8, 2015. http://4www.goldennumber.net/parthenon-phi-golden-ratio/.

"Phidias." AncientGreece.com. 2012. Accessed May 13, 2015. http://www.ancientgreece.com/s/People/Phidias/.5

Julie Rehmeyer. "Still Debating with Plato." ScienceNews.org. April 25, 2008. Accessed April 11, 2015. https://www.sciencenews.org/article/6still-debating-plato

H. E. Huntley. The Divine Proportion: A Study in Mathematical Beauty,. New York, New York: Dover Publications, 1970.7

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see the number phi as a recurrent number . This is evidence of the intentional use of the golden 8

ratio in aesthetics as it relates to architecture. The first written historical piece of evidence of this

theory however, is stated in Euclid’s Elements.

Euclid, perhaps the greatest figure in the history of mathematics, referred to, in his

publication, Elements, a “dividing a line in the extreme and mean ratio” at the 1.618 point. This

reference to a mean led to the re-naming of the golden ratio to the golden mean . 9

Most Greek temples subsequently, followed these rules of mathematical aesthetics. The

analysis of the Parthenon my be applied to many other Greek temples that were built at the time.

Fibonacci Solidifying the Connection Between Math and Nature:

Plato’s belief that the golden ratio was derived from nature was backed up by Leonardo

Fibonacci in the twelfth century when the mathematician discovered a recurrent mathematic

sequence that was present in nature. This sequence of numbers is called the Fibonacci sequence

as it was discovered by its namesake. Fibonacci discovered this sequence while investigating

the reproduction of rabbits and discovered a sequence in the formation of the family tree of the

rabbits which was then named after him . 10

The fibonacci sequence is made up of numbers that are all the addition of the previous

two numbers in the sequence. The sequence starts with 0 and 1 and continues with 1, 2, 3, 5, 8,

Paul Calter. "Polygons, Tilings, & Sacred Geometry." <i>Drathmouth.edu</i>. Dartmouth College, 1 Jan. 1998. Accessed Apr 7, 2015.8

Ibid.9

Alfred Posamentier and Lehmann Ingmar. "The Fibonacci Numbers in Nature." In The Fabulous Fibonacci Numbers, 59 - 76. Amherst, New 10

York: Prometheus Books, 2007.

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13, 21, 34 and continues in this fashion. The sequence is infinite as one can continue adding the

previous numbers forever, in theory. For example, we can continue the sequence with 55, 89,

144, 233, 377, 610, 987, 1597, 2584, 4181. The sequence is represented by the following

formulae: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. “F” being the first term in the

sequence and “n” being the interchangeable Fibonacci number . 11

The Fibonacci sequence is present everywhere one looks in nature. From family trees to

body proportions to the positioning of plant leaves around a stem, we can see this sequence

emerge from the seeming chaos of nature.

The branching of trees is a proof of the presence of mathematics in nature. Each level at

which a tree branches off, the number of branches will increase by one fibonacci number. This is

shown in Figure 3.

Pinecones are another example of an organism that exhibits fibonacci qualities. On a

pinecone, there are two discernible spiral directions that start from one pole of the pinecone and

end at the other pole. The amount of spirals going clockwise and the number of spirals that go

counterclockwise will always be two adjacent numbers in the fibonacci sequence. For example,

the pinecone is Figure 4 shows that the spirals go clockwise, indicated by the green lines, are 13

Sloane. "Fibonacci Sequence." Oeis.org. April 30, 1991. Accessed May 7, 2015. http://oeis.org/A000045.11

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and the spirals that go counterclockwise, indicated by the red lines, are 8. 13 is the number that

follows 8 in the fibonacci sequence . 12

The presence of this sequence in nature is most commonly illustrated in the positioning of

the leaves on a fern plant. These leaves grow in a spiral pattern to optimize the surface area of

the leaves that are exposed to sunlight. The sequence allows for one leaf to never cast shade on

another leaf as it would not maximize the surface area of the leaf in the sun. This idea is

illustrated by Figure 5. This spiraling of the plant leaves is called phyllotaxis. The phyllotaxis

creates a numerical rhythm of 1.618.

This brings us back to the golden mean and demonstrates its relation to the Fibonacci

sequence . When we take any two adjacent numbers in the fibonacci sequence and use them as 13

measurements for the sides of rectangles we can see the golden ratio emerge again. Any

rectangle constructed from two adjacent fibonacci numbers will have the height to width ratio of

1:1.618. This rectangle is called the golden rectangle, as the are heavily related to the golden

ratio . 14

Fetchner and the Golden Rectangle:

Posamentier, Alfred S, and Ingmar Lehmann. "The Fibonacci Numbers in Nature." In The Fabulous Fibonacci Numbers, 59 - 76. Amherst, 12New York: Prometheus Books, 2007.

Ibid. 59 - 76.13

Alfred Posamentier and Lehmann Ingmar. “The Fibonacci Numbers and the Golden Ratio." In The Fabulous Fibonacci Numbers, 107 - 116. 14Amherst, New York: Prometheus Books, 2007.

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In 1876, a psychologist, Gustav Fechner carried out a psychological study to find what

proportions of a rectangle were the most aesthetically pleasing to the human eye. Fechner tested

228 men and 119 women. He handed the test subjects a paper with a selection of ten differently

proportioned rectangles . 15

The first rectangle had the ratio of 1:1, the second one had a ratio of 5:6, the third had the

ratio of 4:5, the fourth had the ratio of 3:4, the fifth rectangle’s ratio was 20:29, the sixth

rectangle was 2:3, the seventh one was the fibonacci derived rectangle with the ratio of 21:34,

the eighth rectangle had the ratio of 13:23, the ninth one had a ratio of 1:2 and the final rectangle

had a ratio of 2:5. The rectangles are shown in Figure 6.

When he asked what the most visually pleasing and least visually pleasing rectangles on

the page were, Fechner found that the rectangle that was most popularly picked as the most

visually pleasing was the fibonacci rectangle with the ratio of 21:34 or 1:1.618. 35 percent of the

test subjects chose the golden rectangle above all others and none of them chose it as the least

aesthetically pleasing . 16

This psychological experiment had opened up the possibility to use these proportions in

aesthetics for mainstream architecture and art. The Fechner experiment gave rise to many new

possibilities in architecture.

A Contemporary Example of the Golden Rectangle in Architecture:

Ibid. 107 - 11615

Ibid. 107 - 11616

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In 1945, famed architect, Le Corbusier, finished constructing a building entitled Untié

d’Habitation . Le Corbusier was intrigued in the golden ratio and the golden rectangle. He 17

wanted to build a living space that was optimal for the human proportions. He then developed

the Modulor which is a concept that takes Leonardo DaVinci’s Vitruvian man further. The

Modulor man breaks the human body into golden sections using with the fibonacci spiral and

golden rectangles. By using this perfectly calculated man, Le Corbusier was able to create

“mathematically ideal dimensions for all construction” . 18

In Le Corbusier’s Unité d’Habitation, we can see many instances where there is an

application of the golden rectangles throughout the building. For example, in figure 7 we can see

that the balconies are divided into uneven sections that are equal to the golden mean. We can

also see another instance of where the building shows fibonacci proportions in figure 8 . 19

It is interesting how the Unité d’Habitation was intended to be the mathematical ideal for

the human proportions. It is not surprising that the human eye would be more prone to desiring

proportions and sequences that mimic the ones found in nature. It is hard-wired into the human

brain to find this aesthetically pleasing as our primitive ancestors were once so in-tune with the

nature that surrounded them.

"Unité D'habitation, Marseille, France, 1945." Fondationlecorbusier.fr. Accessed May 9, 2015. http://www.fondationlecorbusier.fr/corbuweb/17morpheus.aspx?sysId=13&IrisObjectId=5234&sysLanguage=en-en&itemPos=58&itemCount=78&sysParentId=64&sysParentName=home.

Mahajan, Aniket. "Le Modulor." Slideshare. March 28, 2014. Accessed May 3, 2015.18

Ibid.19

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The fibonacci sequence has proven the existence of a deliberate spacial organization in

nature. This spacial logic has been utilized in architectural design to this day.

When analyzing the fibonacci sequence and its relation to the world we are able to

understand a sense of emergence where nature is the origin of and inspires the structure of

everything. We are able to see that mathematics have emerged from certain geometries found in

the seemingly chaotic and disorganized natural entities on Earth. And from these geometries, we

see the emergence of patterns that have been used in architecture throughout many time periods,

finds its beginnings in ancient Greek architecture but still very present in modern architecture

found everywhere today.

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Figures Cited

Fig. 1. Meisner, Gary. PhiMatrix Analysis of Parthenon. January 20, 2013. The Golden Number.

May 1, 2015. http://www.goldennumber.net/parthenon-phi-golden-ratio/.

Fig. 2. Meisner, Gary. PhiMatrix Analysis of Parthenon. January 20, 2013. The Golden Number.

May 1, 2015. http://www.goldennumber.net/parthenon-phi-golden-ratio/.

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Fig. 3. Knott, Ron. Fibonacci Family Tree. 30 October, 2010. Surrey. University of Surrey, 22

Sept. 2010. April 4, 2015.

Fig. 4. Parveen, Nikhat. Fibonacci In the Pinecone. University of Geogia. April 8, 2015.

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Fig. 5. Parveen, Nikhat. Phyllotaxis in Plants . University of Geogia. April 8, 2015.

Fig. 6. Posamentier, Alfred S, and Ingmar Lehmann. Fechner’s Rectangles. 2007. In The

Fabulous Fibonacci Numbers. Amherst, New York: Prometheus Books, 2007.

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Fig. 7. Mahajan, Aniket. “Untié d’Habitation Fibonacci Balconies." March 28, 2014. Slidshare.

May 3, 2015.

Fig. 8. Mahajan, Aniket. “Untié d’Habitation Using Fibonacci Arrangement." March 28, 2014.

Slidshare. May 3, 2015.

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Works Cited

"Ancient Greece: The Parthenon." British Museum. Accessed Apr 12, 2015. https://

www.britishmuseum.org/PDF/Visit_Greece_Parthenon_KS2.pdf.

Calter, Paul. "Polygons, Tilings, & Sacred Geometry." <i>Drathmouth.edu</i>. Dartmouth

College, 1 Jan. 1998. Accessed Apr 7, 2015.

Huntley, H. E. The Divine Proportion: A Study in Mathematical Beauty,. New York, New York:

Dover Publications, 1970.

Mahajan, Aniket. "Le Modulor." Slideshare. March 28, 2014. Accessed May 3, 2015.

Meisner, Gary. "The Parthenon and Phi, the Golden Ratio." TheGoldenNumber.net. January 20,

2013. Accessed May 8, 2015. http://www.goldennumber.net/parthenon-phi-golden-ratio/.

"Phidias." Ancient Greece. 2012. Accessed May 13, 2015. http://www.ancientgreece.com/s/

People/Phidias/.

Posamentier, Alfred S, and Ingmar Lehmann. “The Fibonacci Numbers and the Golden Ratio." In

The Fabulous Fibonacci Numbers, 107 - 116. Amherst, New York: Prometheus Books, 2007.

Posamentier, Alfred S, and Ingmar Lehmann. "The Fibonacci Numbers in Nature." In The

Fabulous Fibonacci Numbers, 59 - 76. Amherst, New York: Prometheus Books, 2007.

Rehmeyer, Julie. "Still Debating with Plato." Science News. April 25, 2008. Accessed April 11,

2015.

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http://www.ancientgreece.com/s/People/Phidias/

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